Virtual Mosaic Knots

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Virtual Mosaic Knots
Irina T. Garduño
University of Illinois at Chicago
[email protected]
October 24, 2009
Dedicated to my mother, Maria E. Luna, and my father,
Enrique Garduño.
Acknowledgments
This research was sponsored by the Mathematical Association of America,
funded by the National Security Agency and the National Science
Foundation, and was made possible by McKendree University under the
direction of Dr. J. Alan Alewine and Dr. Heather A. Dye.
Abstract
We extend mosaic knot theory to virtual knots and define a new
type of knot: virtual mosaic knot. As in classical knots, Reidemeister
moves are applied to a virtual mosaic knot to transform one knot
diagram into another. Additionally, given the mosaic number of a
virtual knot, we find an upper bound on the sum of the classical and
virtual crossing numbers. Furthermore, given the classical and virtual
crossing numbers of a knot, we find a lower bound on the virtual
mosaic number of a knot.
1
Introduction
In the nineteenth century, Johann Carl Friedrich Gauss (1777-1855) began
to mathematically study knots [4]. Gauss showed in 1833 that the linking
1
number of two knots can be computed by an integral. In 1867, William
Thomson (Lord Kelvin) (1824-1907) [4] presented his vortex-atom theory,
which states that each chemical element has its own knot form and can be
differentiated by their given knot form [4]. Although Thomson’s theory was
misguided, Peter Guthrie Tait (1831-1901) created the first knot tables in
1876. Since then, knot theory has expanded and has become a major part of
the mathematics field called topology.
Vaughan Jones discovered the Jones polynomial in 1983 and in 1990 he
won the Fields Medal for his contribution in the field [6], inspiring substantial
activity in knot theory. In 1999, Louis H. Kauffman defined virtual knots
[1]. Almost ten years later mosaic knots were defined by Samuel J. Lomonaco
and Louis H. Kauffman [2].
In Section 2, we provide background information and definitions. We
begin with classical knots and Reidemeisters theorem. Next, we recall virtual
knot theory and mosaic knots. In Section 3, we introduce and describe virtual
mosaic knots. In Section 4, we discuss the lower bound of a mosaic number.
We also show that these bounds can be applied to virtual mosaic knots.
2
2.1
Background
Classical Knots
A knot is an embedding (an injective map) of a circle, S 1 , into R3 . A knot
diagram is a projection of a knot onto the plane in which crossing information
is preferred. Figure 1 is an example of a knot diagram.
Figure 1: A Knot Diagram
We can deform one knot diagram to another by way of stretching and
bending. This process is called planar isotopy, as shown in Figure 2.
Reidemeister moves involve three different types of moves applied to knot
diagrams. We can apply these moves to reduce the number of crossings and
2
Figure 2: Example of Planar Isotopy
to transform a given knot diagram into a simpler yet equivalent knot diagram.
The Reidemeister moves are shown in Figure 3.
Figure 3: Reidemeister Moves
Note that these three moves are actually eight moves after accounting for
over and under crossings.
Figure 4 is an example of a knot diagram transformed into another knot
diagram using Reidemeister moves. Reidemeister proved these moves describe all possible three-dimensional moves, as stated in Theorem 2.1.
Theorem 2.1 (Reidemeister). Two knot diagrams represent the same knot
if and only if one can be transformed into the other by a finite sequence of
Reidemeister moves and planar isotopy moves [3].
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Figure 4: Example of Reidemeister Moves
Figure 5: Equivalence Relation of Two Knots Using Reidemeister Moves
Thus, two knot diagrams are related if their diagrams are associated by
a finite sequence of Reidemeister moves and/or planar isotopy moves. Reidemeister moves allow us to describe the three-dimensional motions of knots
as diagrams. Furthermore, Reidemeister moves allow us to place knots in
equivalence classes.
Recall that an equivalence relation, ≡, on a set X must satisfy the following for all x, y, and z in X:
• Reflexive: x ≡ x
• Symmetry: if x ≡ y, then y ≡ x
• Transitive: if x ≡ y and y ≡ z, then x ≡ z
This idea can be applied to knots. For example, any two diagrams of
the unknot related by the Reidemeister moves are members of the equivalent
class of the unknot. An equivalence class of the element y ∈ X is the set
{x ∈ X : x ≡ y}.
In Figure 6 we see several elements of the equivalence class containing
the unknot. The knot diagrams are equivalent because they are related
by a sequence of Reidemeister moves. That is, Reidemeister moves form
an equivalence relation and the knot diagrams are elements of the same
equivalence class. In this sense we can define a knot to be an equivalence
class of knot diagrams.
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Figure 6: Equivalence Class of Knot Diagrams
2.2
Virtual Knots
Virtual knot diagrams are an extension of classical knot diagrams [1]. Classical and virtual knots are represented in diagram form, and there are an
extended set of Reidemeister moves and planar isotopy moves. The major
difference between virtual and classical knot diagrams is the type of crossings.
A virtual knot diagram is a mapping of S 1 into R2 with classical crossings
and an additional type of crossing called a virtual crossing.
Figure 7: A Classical Crossing, a Virtual Crossing, and a Virtual Knot Diagram.
Notice in Figure 7 that the virtual crossing of a knot diagram does not
include over/under markings.
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Figure 8: Virtual Reidemeister Moves
Figure 9: Example of Virtual Reidemeister Moves
In virtual knot theory there are four additional Reidemeister moves. The
first three virtual Reidemeister moves are analogous to the classical Reidemeister moves. The fourth move involves both types of crossings as shown
in Figure 8. Two virtual knot diagrams are equivalent if their diagrams are
related by a finite sequence of virtual and classical Reidemeister moves and
planar isotopy.
We see two equivalent virtual knot diagrams in Figure 9 and can now
define a virtual knot [1]. A virtual knot is an equivalence class of virtual
knot diagrams determined by all the Reidemeister moves.
6
2.3
Knot Invariant
We now introduce the idea of an invariant. A knot invariant is a numerical
quantity or a property associated with knot diagrams where the Reidemeister
moves do not change the numerical quantity or the property. The crossing
number of a knot is the minimum number of classical crossings in any equivalent knot diagram.
Figure 10: Classical Crossing Number
Figure 10 shows two different yet equivalent knot diagrams related by
Reidemeister moves. Notice that the number of crossings decreases with
each Reidemeiste move. Therefore, the crossing number is zero because we
want the least number of crossings in any of its equivalent knot diagrams.
Figure 11: Mosaic Tiles, and a 5-mosaic
In the same manner we can define the virtual crossing number. A virtual
crossing number of a knot is the minimum number of virtual crossings in any
equivalent virtual knot diagram. We can also define the complete crossing
number as the sum of the virtual and classical crossing numbers.
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Figure 12: Connection Points
2.4
Mosaic Knots
We now introduce the concept of mosaic knots [2].
Remark 2.1. The only difference between classical knots and mosaic knots
is that mosaic knot diagrams are laid on tiles.
Figure 13: Contiguous Tiles
An n-mosaic is an n × n matrix composed of mosaic tiles, as shown in
Figure 11.
A connection point is the end point of any curve or line within a tile and
located at the midpoint of a tile’s edge. These connection points allow us
to create n-mosaic knots. A knot n-mosaic is a mosaic with all of its tiles
joined at the connection points.
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Figure 14: Suitably Connected Tiles and Not Suitably Connected Tiles
Contiguous tiles, as shown in Figure 13, are two tiles in a mosaic diagram
that are placed immediately next to each other.
Furthermore, two tiles are suitably connected tiles when all of the connection points of a tile touch the connection points of a contiguous tile in a
mosaic, as shown in Figure 14.
A mosaic number is the smallest integer for which a knot is representable
as a knot n-mosaic. Figure 15 exhibits the 11 mosaic planar isotopy moves.
Remark 2.2. We can use these moves to obtain a diagram with the mosaic
number. Furthermore, we reduce a column and a row using mosaic planar
isotopy moves if there are no crossings or corners involved in the row or
column (prove this in section 4).
Notice that in Figure 16 the left hand knot is a 6×6 mosaic knot diagram
that is reduced to a 5 × 5 mosaic knot diagram.
In Figure 18, the left hand knot is a 7 × 7 mosaic knot diagram that is
reduced to a 6 × 6 mosaic knot diagram and finally to a 5 × 5 mosaic knot
diagram. Moreover, we see that the left hand knot is equivalent to the right
hand knot. Hence, two mosaic knot diagrams are equivalent if one can be
transformed into the other by a finite sequence of mosaic Reidemeister moves
and/or mosaic planar isotopy.
Notice that in both the mosaic planar isotopy and mosaic Reidemeister
moves we have tiles with dotted lines. These tiles are called non-deterministic
tiles. A non-deterministic tile is a tile that denotes two possible outcomes.
For example, in Figure 17 we see that the mosaic Reidemeister moves have
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Figure 15: Mosaic Planar Isotopy
Figure 16: Example of Mosaic Planar Isotopy
dotted lines that go in two directions. In these cases, the knot continues in
one of the two directions. The same holds for a tile that has dotted lines in
only one direction, as shown in Figure 19.
3
Virtual Mosaic Knots
We extend mosaic knot theory and introduce virtual mosaic knot theory.
Recall that n-mosaics are composed of mosaic tiles. Virtual mosaic tiles are
a collection of mosaic tiles that include the virtual mosaic tile.
A virtual n-mosaic is an n × n matrix of mosaic tiles that potentially
includes the virtual tile, as shown in Figure 20. We can now define a knot
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Figure 17: Mosaic Reidemeister Moves
virtual mosaic. A virtual n-mosaic knot is a virtual n-mosaic where all of its
tiles are suitably connected [Figure21b].
Remark 3.1. Virtual mosaic knot diagrams do not have specification on the
matrix size.
A virtual mosaic number is the smallest integer m for which a virtual
mosaic knot is representable as a virtual n-mosaic knot.
We extend mosaic planar isotopy to virtual mosaic planar isotopy by
adding two virtual mosaic planar isotopy moves, P12 and P13 shown in
Figure 22, and four virtual mosaic Reidemeister moves shown in Figure 23.
Figure 18: Example of Mosaic Reidemeister Moves
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Figure 19: Non-Deterministic Tiles
Figure 20: Virtual Mosaic Tile
In the following example, Figure 24, we apply a virtual mosaic Reidemeister move 2 so that the left hand knot is equivalent to the right hand
knot.
Two virtual n-mosaic knot diagrams are equivalent if one can be transformed into the other by a finite sequence of classical and virtual mosaic Reidemeister moves and/or classical and virtual mosaic planar isotopy moves.
4
4.1
Lower and Upper Bounds of a Mosaic
Upper Bound on The Crossing Number
Proposition 4.1. The edges of a virtual n-mosaic knot cannot contain classical crossings or virtual crossings.
Proof Suppose we have a virtual n-mosaic knot with virtual mosaic number m. By definition a virtual n-mosaic knot is a virtual mosaic with all
tiles suitably connected. A virtual n-mosaic knot holding a crossing in an
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Figure 21: A Virtual 5-Mosaic and a Knot Virtual 5-Mosaic
Figure 22: Virtual Mosaic Planar Isotopy Moves
edge tile is not suitably connected. So, the edges of a virtual n-mosaic knot
cannot hold crossings.
For example,in Figure 25a we have a suitably connected virtual 5-mosaic.
Notice that none of the crossings are at the edges. The not suitably connected
virtual 5-mosaic in Figure 25b has crossings at the edges that do not ’lead’
to any connection points. This is an example where all the crossings must
not be at the edges. Additionally, all crossings must be inside the 5-mosaic
because if there were any crossings at the edges of the 5-mosaic, then the
5-mosaic will not be suitably connected.
Proposition 4.2. Given a virtual n-mosaic knot, we can reduce a column
and a row using virtual mosaic planar isotopy moves and mosaic Reidemeis13
Figure 23: Virtual Mosaic Reidemeister Moves
ter moves if there are no crossings or corners involved in the column or row.
Proof Suppose we have a virtual n-mosaic knot diagram. By definition
two virtual mosaic knot diagrams are equivalent if one can be transformed
into the other by a finite sequence of classical and virtual mosaic Reidemeister
moves and/or classical and virtual mosaic planar isotopy moves. Since all
the boundary rows or columns have no crossings or corners which could be
reduced via a sequence of mosaic Reidemeister moves and/or virtual mosaic
planar isotopy moves, they represent a different yet equivalent virtual n-
Figure 24: Example of Virtual Mosaic Reidemeister Moves
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Figure 25: Suitably Connected Virtual 5-Mosaic and a Not Suitably Connected Virtual 5-Mosaic
mosaic knot diagram.
Theorem 4.3. Given a virtual knot, let m denote the mosaic number, n denote the classical crossing number, and r denote the virtual crossing number.
Then l ≤ (m − 2)2 , where l = n + r and m ≥ 2.
Proof Suppose we have a virtual mosaic knot with l crossings and mosaic
number m. Now, a virtual n-mosaic knot can be reduced to a n-mosaic
knot which cannot contain crossings at the tile’s edge. Thus all classical and
virtual mosaic crossings are within the m − 2 submatrix. Hence l ≤ (m − 2)2
is an upper bound on l.
Figure 26: A Virtual 4-Mosaic With Alternating Crossings
For example, Figure 27a shows a virtual 5-mosaic with alternating crossings. As noted, all of the crossings are within the walls of the edges. If we
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Figure 27: A Virtual 5-Mosaic with Alternating Crossings
connect the crossings in such a way as to create a knot, we get the following
two knots as shown in Figure 27b and c.
4.2
Lower Bound on The Mosaic Number
Recall Proposition 4.1 and Proposition 4.2.
Theorem 4.4. Given a virtual knot, let m denote the mosaic number, n denote the classical
l√ m crossing number, and r denote the virtual crossing number.
l + 2, where l = n + r and l ≥ 0.
Then m ≥
Proof Suppose we have a virtual mosaic knot with l crossings and mosaic number m. Since there is at most one crossing in each tile, the smallest square-matrix
that could contain all classical and virtual crossings is an
l
√ m l√ m
l ×
l submatrix and since there cannot be any crossings on the edge
l√ m
l + 2.
tiles, there must be two additional rows and columns. Thus, m ≥
Remark 4.1. Theorem 4.3 is a Corollary of Theorem 4.4.
For example, in Figure 28a we see a 2-submosaic enclosed in a virtual
4-mosaic knot diagram. Notice that since there cannot be any classical or
virtual crossings at a tile’s edge, all of the crossings are within the the 2 × 2
submatrix. Although the number of crossings are different in the virtual
5-mosaic knot diagrams shown in Figures 28b and c, both the two virtual
mosaic knot diagrams have a 3-submosaic enclosed in a virtual 5-mosaic knot
diagram where all of the crossings are within the 3 × 3 submatrix.
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Figure 28: A Virtual 4-Mosaic and Virtual 5-Mosaics
5
Conjecture
We conclude this paper with some conjectures that we plan to investigate
in the future. Recall Theorem 4.3. The upper bound on the total crossing
number l of a virtual n-mosaic knot given the mosaic number m can be
reduced to:
• (m − 2)2 − 2 if m is odd
• (m − 2)2 − 1 if m is even, where m ≥ 2
As shown in Figure 27, two corners of the virtual 5-mosaic knot diagram
can undergo virtual mosaic Reidemeister move 1 reducing the crossing number by two. This can be applied to any odd virtual n-mosaic knot as we will
obtain two corners that will undergo classical or virtual mosaic Reidemeister
move 1.
Furthermore, when we pack a virtual 4-mosaic with crossings, as in Figure
26b, we get a link, which is non-empty union of a finite number of disjoint
knots. Since we are dealing with knots and knot diagrams, we must avoid the
use of a link. Thus we must remove at least one crossing to avoid obtaining
a link.
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References
[1] Kauffman, Louis H. Virtual knot theory. European J. Combin. 20 (1999),
no. 7, 663–690. (Reviewer: Olivier Collin).
[2] Lomonaco, Samuel J., and Louis H. Kauffman, Quantum Knots and
Mosaics, Journal of Quantum Information Processing, Vol. 7, Nos. 2-3,
(2008), pp. 85 - 115.
[3] Reidemeister, K. Knot theory. Translated from the German by Leo F.
Boron, Charles O. Christenson and Bryan A. Smith. BCS Associates,
Moscow, Idaho, 1983.
[4] Silver, Daniel, Scottish Physics and Knot Theory’s Odd Origins,
http://www.americanscientist.org/issues/feature/knot-theorys-oddorigins, 2006.
[5] Vick, James W. Topology now!. By Robert Messer and Philip Straffin,
Introduction to topology: pure and applied. By Colin Adams and Robert
Franzosa, Pearson Prentice Hall, Lebanon, IN, 2008.
[6] Weisstein, Eric, Fields Medal, http://mathworld.wolfram.com/FieldsMedal.html,
2009.
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